TUTORIAL REVIEW
www.rsc.org/loc | Lab on a Chip
Basic principles of electrolyte chemistry for microfluidic electrokinetics.
Part II:† Coupling between ion mobility, electrolysis, and acid–base
equilibria‡
Alexandre Persat,x Matthew E. Sussx and Juan G. Santiago*
Received 1st April 2009, Accepted 4th June 2009
First published as an Advance Article on the web 7th July 2009
DOI: 10.1039/b906468k
We present elements of electrolyte dynamics and electrochemistry relevant to microfluidic
electrokinetics experiments. In Part I of this two-paper series, we presented a review and introduction to
the fundamentals of acid–base chemistry. Here, we first summarize the coupling between acid–base
equilibrium chemistry and electrophoretic mobilities of electrolytes, at both infinite and finite dilution.
We then discuss the effects of electrode reactions on microfluidic electrokinetic experiments and derive
a model for pH changes in microchip reservoirs during typical direct-current electrokinetic
experiments. We present a model for the potential drop in typical microchip electrophoresis device. The
latter includes finite element simulation to estimate the relative effects of channel and reservoir
dimensions. Finally, we summarize effects of electrode and electrolyte characteristics on potential drop
in microfluidic devices. As a whole, the discussions highlight the importance of the coupling between
electromigration and electrophoresis, acid–base equilibria, and electrochemical reactions.
Introduction
Electrode chemistry, acid–base chemical equilibria, and ionic
strength have a significant effect on the performance of microfluidic electrokinetic devices. Among other effects, these
Department of Mechanical Engineering, Stanford University, Stanford,
CA, 94305, USA. E-mail: juan.santiago@stanford.edu; Fax: +1 650723-5689; Tel: +1 650-723-7657
† For Part I see ref. 7.
‡ Part of a Lab on a Chip themed issue on fundamental principles and
techniques in microfluidics, edited by Juan Santiago and Chuan-Hua
Chen.
x The first two authors contributed equally to this work.
Alexandre Persat is a PhD
candidate at Stanford University, under the supervision of
Prof. Juan G. Santiago. He
received his MS in Chemical
Engineering
at
Stanford
University, and his Diplome
d’Ing
enieur from l’Ecole Polytechnique, France.
2454 | Lab Chip, 2009, 9, 2454–2469
characteristics determine local electroosmotic flow mobilities,1,2
shapes of electropherograms,2 conductivity and power
consumption,3 and compatibility with chemical and biological
species.4 In typical direct current (DC) electrokinetic (EK)
systems, passive DC electrodes are inserted into end-channel
reservoirs which have characteristic dimensions relatively large
compared to those of microchannels.5,6 Faradaic current (electron transfer to and from electrodes) and ionic (Ohmic) current
together sustain an electric field in the bulk of the electrolyte. The
rate and nature of electrolytic reactions can strongly influence the
acid–base equilibria (including pH) at channel reservoirs and in
the channel itself. In turn, these acid–base equilibria determine
local ion mobilities and conductivities of the electrolyte, and so
Matthew E. Suss received his B.
Eng degree in Mechanical
Engineering
from
McGill
University, Montreal, in 2007.
Currently, he is finishing
a M.Sc. at Stanford University,
Stanford, CA where his research
has been supported by an
FQRNT fellowship from the
Quebec government. His work in
the Stanford Microfluidics
Laboratory focuses on the
development of highly efficient
electroosmotic micropumps for
drug delivery applications.
Matthew additionally spent two summers as an undergraduate
research assistant working on three-dimensional characterization
of osseous implant surface topography.
This journal is ª The Royal Society of Chemistry 2009
reactions occurring at the electrode surfaces are intimately
coupled to local chemistry and electrokinetic phenomena.
In Part I of this two-paper series,7 we presented elements of
acid–base equilibrium chemistry and pH buffering, and we discussed their importance to microfluidic electrokinetic experiments. In this second paper, we present an introduction to the
basic principles of the coupling between electrolyte mobilities,
acid–base equilibria, and electrolysis in typical DC EK systems.
We first present a summary of the effects of ionic strength and
pH on (observable) effective electrophoretic mobilities. As with
Part I, we first summarize dilute approximations and then discuss
extrapolation to finite ionic strength. We characterize conductivity in terms of these effective mobilities. We then summarize
the coupling between ionic conductivity, system geometry, and
electrolytic reaction rates. We analyze typical electrolyte systems
to demonstrate and summarize the effects of electrolysis on endchannel pH values. We additionally analyze electrolytic and
Ohmic sources of potential loss, which may aid in design of EK
systems. Our intent is to provide an introductory review of
principles, provide citations for key phenomena, and summarize
various illustrative analyses.
Nonemclature
e
cX
cX,z
d
D
Eeq
electron
analytical concentration of X
concentration of X in valence state z (M)
Microchannel diameter (m)
Reservoir diameter (m)
Equilibrium potential (V)
Prof. Juan G. Santiago is an
Associate Professor in the
Mechanical
Engineering
Department at Stanford, and
specializes in microscale transport phenomena and electrokinetics. He received his MS and
PhD in Mechanical Engineering
from the University of Illinois at
Urbana-Champaign (UIUC).
His research includes the development of microsystems for onchip
electrophoresis,
drug
delivery, sample concentration
methods, and miniature fuel
cells. He has received a Frederick Emmons Terman Faculty
Fellowship (’98–’01); won the National Inventor’s Hall of Fame
Collegiate Inventors Competition (’01); was awarded the
Outstanding Achievement in Academia Award by the GEM
Foundation (’06); and was awarded a National Science Foundation Presidential Early Career Award for Scientists and Engineers
(PECASE) (’03–’08). Santiago has given 13 keynote and named
lectures and more than 100 additional invited lectures. He and his
students have been awarded nine best paper and best poster awards.
Since 1998, he has graduated 14 PhD students and advised 10
postdoctoral researchers. He has authored and co-authored 95
archival publications, authored and co-authored 190 conference
papers, and been awarded 25 patents.
This journal is ª The Royal Society of Chemistry 2009
F
f
gA,z
i
I
jo,a
Ja/c
X
Ka
L
Leff
Seff
S
nX
pX
Vapp
Vc
Vres
g
gX
g
f
mX
moX,z
mN
X,z
ha/c
s
Y
Faraday’s constant x 96485 C mol1
Transport (or transference) number
Fraction of A in valence state z
Current (A)
Ionic strength (M)
Anode exchange current density (A m2)
Flux of X in anode/cathode reservoir
Acid dissociation constant
Channel length (m)
Effective length of reservoir (m)
Effective cross sectional area of reservoir (m2)
Channel cross sectional area (m2)
Minimum valence of species X
Maximum valence of species X
Applied potential (V)
Potential across microchannel (V)
Potential drop across reservoir (V)
Ratio of anode to cathode reservoir
conductivities
Activity coefficient of species X
Mean activity coefficient of a cation/anion pair
Local potential (V)
Effective electrophoretic mobility of X (m2
V1 s1)
Fully ionized mobility of Xz (m2 V1 s1)
Mobility of Xz at zero ionic strength (m2 V1
s1)
Anode/cathode overpotential (V)
Conductivity (S m1)
Reservoir volume (m3)
Effective mobility
In the Background section of Part I, we presented fairly general
expressions for chemical equilibrium based on activity coefficients. We recall that activity coefficients represent the link
between experimentally accessible quantities (i.e., species
concentrations) and the energetics of equilibrium reactions (i.e.,
through chemical potentials).8 In particular, we noted activity
coefficients strongly depend on ionic strength. We also presented
a general treatment of acid–base equilibrium reactions, and
derived general expressions for concentrations of species in their
various protonated states, as a function of pH and dissociation
constants. In this section, we will first discuss the effects of acid–
base equilibrium on electrophoretic transport coefficients of
species. We will then relax the assumption of ideal dilute solutions and summarize the effects of ionic strength on ionic
transport.
Effective mobility of a partially ionized weak electrolyte ion
In this section, we present relations applicable to the electrophoretic transport of multi-ion systems. The discussion applies to
both weak and strong electrolytes and is crucial to the control
and interpretation of experiments and measurements which
depend on ion mobility. Such applications include capillary zone
electrophoresis, isotachophoresis, isoelectric focusing, field
Lab Chip, 2009, 9, 2454–2469 | 2455
amplified sample stacking, and a wide variety of other electrokinetic techniques.9,10 We here restrict our discussion to electromigration in free solution (e.g., versus electromigration of
macromolecules through gels).
We here define ion mobility as in most electrophoresis applications as the proportionality constant between observed ion
velocity and electric field (a signed quantity).11 Electrophoretic
mobility is typically quantified by capillary electrophoresis.9 The
time-averaged mobility of a species in solution can be related to
the fraction of time the species spends in its various relevant
ionization states. For observation times much longer than the
time scales associated with proton exchange reactions (typically
very short11), we observe an ‘‘effective mobility’’ which is the time
average of the fully ionized mobilities associated with each
ionization state. For a simple monovalent weak acid, the
mobility of the species X of analytical concentration cX ¼ cX,0 +
cX,1 at a given pH is then:12
o
mX ¼ aXmX,1.
(1)
Here aX ¼ cX,1/cXis the degree of dissociation of the acid X,
and moX,1 is the fully ionized mobility of X (the superscript o
indicates an ideal state where the ion is always charged at the
valence described by the second subscript{). Our notation
follows the conventions we established in Part I; so cX,1 and
cX are respectively equal to the concentration of species X in the
z ¼ 1 valence state and the total concentration of X (summed
over all ionization states). More generally, for species X with
several ionization states, the motion in response to an electric
field is described as follows:
!
pX
X
o
~¼
~
~
u ¼ mX E
(2)
m gX;z E;
X;z
z¼nX
where E~
p is the local applied electric vector field and u~
p is the
electrophoretic velocity vector of the species. This velocity
describes the time-averaged transport associated with all
valences of the species family X (here ‘‘family’’ includes all
ionization states of X). mX,zo is the fully ionized mobility corresponding specifically to the charge state z. We define gX,z as the
fraction of X in charge state z:
gX;z h
cX;z
LX;z czH
¼ pX
;
X
cX
z
LX;z cH
(3)
z¼nX
where LX,z was defined in Part I. Note that the degree of dissociation, aX, is the sum of gX,z over all z s 0. Accordingly, the
effective mobility of X is given by:
mX ¼
pX
X
moX;z gX;z :
gX;0 ¼
1
1 þ KX;1 =cH
gX;1 ¼
KX;1 =cH
:
1 þ KX;1 =cH
and
Since the mobility of a neutral species is zero, we have
1
mX ¼ moX;0 gX;0 þ moX;1 gX;1 ¼ moX;1
1
þ
c
=KX;1
H
|{z}
¼0
¼ moX;1
1
1 þ 10pKX;1 pH
(5)
:
Note here the importance of dissociation states on the observed
mobility of a monovalent acid. For example, fully ionized
mobilities of monovalent acids (z ¼ 1) range from roughly
20 109 m2 V1 s1 to 35 109 m2 V1 s1 (e.g., among
the Good’s buffer ions); a range with a maximum-to-minimum
ratio of roughly 2. In contrast, the ratio cH/KX,1 can vary by
more than 2 orders of magnitude in practice; and so the observed
mobility of a monovalent acid varies between 0 and its fully
ionized value.
As a second example, we provide the effective mobility, mX, of
a divalent acid with two pKa’s and corresponding valences of
2 and 1. For this case, LX,0 ¼ 1, LX,1 ¼ KX,2 and
LX,2 ¼ KX,2KX,1; and the effective electrophoretic mobility of
this acid X is
mX ¼
moX;1 þ moX;2 10pHpKX;2
:
1 þ 10pHpKX;2 þ 10pKX;1 pH
(6)
Relations for effective mobilities of other multivalent species,
including mono- and di-valent weak bases, follow easily from
this approach to the formulation.
Back-of-the-envelope estimates for monovalent weak acids and
bases
For a simple acid with a single relevant pK1 (HA # A + H+ ),
inspection of equation (5) above yields the intuitive results that
If pH pK1, mX z 0
If pH zpK1, mX z moX,1/2
(4)
z¼nX
mX is the mobility which is directly observable and quantified
in a uniform pH and uniform ionic strength capillary zone
{ moX,z is directly observable in electrophoresis type experiments only if
there is a pH where the species exists predominantly in the valence
state z (i.e., such that cX,z y cX). For example, direct observation of
a divalent acid mobility value moX,1 is not possible if KX,2 is less than
10 fold lower than KX,1.
2456 | Lab Chip, 2009, 9, 2454–2469
electrophoresis experiment. In the case of a monovalent acid
(with only one relevant pKa ¼ pKX,1) and a fully ionized
mobility moX,1, we obtain LX,0 ¼ 1, LX,1 ¼ KX,1 (see Part I)
so that:
If pH [ pK1, mX z moX,1
Conversely, for weak bases with ionization states 0 (base) and
+1 (acid) and acid dissociation constant pK +1 (BH+ # B + H+ ),
we write similarly
If pH [ pK+1, mX z 0
If pH zpK+1, mX z moX,+1/2
If pH pK+1, mX z moX,+1
This journal is ª The Royal Society of Chemistry 2009
Although very useful as reference points, we caution the reader
not to employ linear interpolations between these points due to
the exponential nature of pH and pK1.
Consider again the example of a pH buffer containing a weak
acid HA and a weak base B. The relevant equilibrium reactions
were HA # A + H+ and BH+ # B + H+. If the pKa
values of these electrolytes are sufficiently far from each
other (say a value of 2 or greater), then the calculations of
mobilities can be simplified. If the buffer is sensibly titrated to,
say, pH ¼ pKHA,1 < pKB,+1 2, then we can assume the base B
is fully ionized. Therefore, cHA,1 y cHA/2 and cB,1 y cB.
Accordingly, the effective mobilities are simply mHA z moHA,1/2
and mB z moB,+1.
Transference number and conductivity
Effective ion mobilities determine time-average electrophoretic
velocities of ions in solutions, their respective contributions to
total current, and overall ionic conductivity. Two important
relations follow from these formulations. First, we define the
ionic conductivity in the usual way as
X X
s¼F
zmoX;z cX;z ;
(7)
z
where F is Faraday’s constant, and cX,z the concentration of X in
valence state z. The summation is performed over all species in
solution (i.e., over all valences of all families). Note that
conductivity is a function of local acid–base equilibria
through cX,z.
As an illustrative example, we summarize here the conductivity
of a simple weak acid buffer HA titrated with a strong base B.
The relevant equilibrium reaction is HA # A + H+. The
conductivity of the solution is then
s ¼ F(moB,+1cB,+1 + moHcH moA,1cA,1 moOHcOH).
(8)
This expression simplifies when assuming that the current
carried by hydronium and hydroxide ions is small compared
to the solutes. Under this assumption (sometimes called the
‘‘safe pH’’ approximationk,13), the conductivity is approximately:
s ¼ F(moB,+1cB, +1 moA,1cA,1),
(9)
where the weak electrolyte ion species concentration obviously
depends on pH. For example, if cA is the weak acid concentration, then:
k The ‘‘safe pH’’ approximation is more restrictive than the ‘‘moderate
pH’’ approximation.7 The electrophoretic mobilities of hydronium and
hydroxide ions are large. Even at small concentrations, these ions can
carry a large portion of the current. Therefore, in the safe pH
approximation, concentrations of hydronium and hydroxide ions need
to be significantly smaller than in the moderate pH approximation. The
safe pH approximation applies to the pH range 5–9 for electrolytes
with concentrations on the order of 10 mM.13
This journal is ª The Royal Society of Chemistry 2009
FcA
o
o
m
:
m
B;þ1
A;1
1 þ 10pKa pH
(10)
A second quantity which we shall leverage below is the
transference number (or transport number) of species X, fX,
defined as:
Back-of-the-envelope calculations for two weak electrolytes in
a buffer
X
s¼
fX ¼
F mX cX
;
s
(11)
where mX is the effective electrophoretic mobility of X, and cX is
the analytical concentration of X. The transference number
indicates the relative contribution of a species to the total
current. Note that, in a binary electrolyte, the transference
number of the cation and anion are not necessarily equal, as these
don’t necessarily have precisely equal effective mobilities. It is
a quantity intertwined with the history of electrophoresis14 as
individual ion contributions to current (and their effective
mobility) can be determined by comparing the conductivity
of electrolytes.8 More importantly, transference number is
particularly useful in displacement processes such as isotachophoresis,15 moving boundary electrophoresis,16,17 and
isoelectric focusing.18,19
In Part I, we saw that pKa is a function of ionic strength
through charge shielding effects. In the next section, we will
discuss how fully-ionized mobility mX,zo is also a function of ionic
strength.
Mobilities at finite dilution: Effect of ionic strength
We here summarize the dependence of electrophoretic mobility
on ionic strength. We also provide several practically useful
equations we can leverage to design and optimize electrokinetic
processes. The effect of ionic strength on electrokinetic experiments (e.g., the shape and interpretation of electropherograms)
can be significant and yet these are often ignored. General
modeling of the phenomena is complex and numerous models
have been developed, many of which require species-specific
empirical or approximate parameters. In our opinion, the best
treatment of this theory can be found in the book by Robinson
and Stokes,8 although there are many and more recent publications.12 Further, in modeling electrokinetic systems, ionic
strength couples with solution composition. This typically
requires iterative resolution schemes (e.g., to solve for conductivity) and always makes analytical models more complex.
P
We recall that ionic strength, I, is defined as I ¼ (1/2) z2cX,z.
In the current context, I can be interpreted as a measure of the
deviation from the infinite dilution approximation. At finite high
ionic strength, each ion is surrounded by a counterion atmosphere with dimensions on the order of the Debye screening
length lD.8 Ionic strength then affects electrophoretic mobility
through two distinct mechanisms. First, there is a hydrodynamic
phenomenon called the electrophoretic effect associated with the
drag force which the (moving) counter ions exert on the respective ion.8 Second is an electrodynamic effect called the relaxation
effect. The latter is associated with the polarization of the
counterion atmosphere which reduces the (local) electric field
experienced by the ion.8 Both effects act to reduce the electrophoretic velocity of the ion.
Lab Chip, 2009, 9, 2454–2469 | 2457
We here suggest a correction for electrophoretic mobilities
which we have found to be a useful trade-off between accuracy
and simplicity. The Pitts equation as suggested by Li et al.12
includes ionic strength and finite-radius effects and is valid for
binary electrolytes. In water at 25 C, the Pitts equation becomes:
2q
o
N
4
N
mX;z ¼ mX;z 3:1 10 z þ 0:39jzjjzci j
pffiffiffi m
1 þ q X;z
pffiffiffi
(12)
I
pffiffiffi;
1 þ 0:33a I
where mN
X is the ion mobility at infinite dilution (i.e., at I ¼ 0)
expressed in equation (12) in cm2 V1 s1, zci is the valence of the
counterion, and a is an effective atomic radius given in Å.
Finding data on effective atomic radii remains difficult for
compounds other than small ions and an obstacle to accurate
mobility prediction. For small ions, there are several relevant
references for parameter a.20 The value of a typically varies
between about 3 and 10 Å. The parameter q depends on the
infinite dilution value of counterion mobility, mN
ci,z, as follows:
q¼
N
mN
jzzci j
X;z þ mci;z
:
N
jzj þ jzci j jzjmX;z þ jzci jmN
ci;z
(13)
For example, for a symmetric (binary) electrolyte, q ¼ 1/2.
Note that the Pitts equation is strictly valid for binary electrolytes. The Fuoss-Onsager correction is more adapted to mixtures
of an arbitrary number of ionic species.21
We can combine equations (12) and (4) to derive the effective
mobility of a weak electrolyte at finite ionic strength. Note that,
in most cases, the concentration ratios gX,z in (4) are functions of
pK’s. pK’s are intimately (and mathematically implicitly) coupled
to ionic strength. That is, ionic strength determines pK values
(e.g., through the Davies equation described in Part I). pK values,
in turn, determine ionization states and therefore ionic strength.
Hence, estimating actual mobility values almost always requires
numerical, iterative solutions. In Part I, we summarize various
computational tools available online to determine acid–base
equilibria and ionic-strength-dependent effective mobilities.
In Fig. 1, we show typical calculations leveraging these two
models. Shown is the electrophoretic mobility of fluorescein,
arguably the most common (and inexpensive) fluorescent dye
used in microfluidics, as a function of pH and increasing values
of ionic strength. The calculations assume that ionic strength and
pH are set by a background electrolyte (so that fluorescein does
not participate in ionic strength or buffering here, and this
greatly simplifies expression for its mobility). We assume the
relevant pK values and mobilities of fluorescein reported by
Whang and Yeung22 and Mchedlov-Petrossyan et al.23 and
Milanova et al.24 The largest variations of mobility are due
largely to the strong effects of ionization states discussed earlier.
Fluorescein has infinite dilution pK values of 4.45 and 6.8 for
valence pairs of (0, 1) and (1, 2), respectively. The respective
values of pK1and pK2 change by about 0.21 and 0.43 pKa units
from 0 to 100 mM ionic strength (respectively, by 0.27 and 0.54
units from 0 to 500 mM). For a fixed pH value, the mobility
decreases with ionic strength as the electrophoretic and relaxation effects discussed above become increasingly more important. Overall, we see that fluorescein mobility is a strong function
2458 | Lab Chip, 2009, 9, 2454–2469
Fig. 1 Ionic strength and pH effects on electrophoretic mobility of
fluorescein. We use the Pitts equation (equation (12)) and Davies equations (eq. (40) of Part I of this two-paper series) to correct electrophoretic
mobility and pKa for ionic strength effects, respectively. We plot the
absolute value of fluorescein mobility (all values are actually negative).
The values of concentrations correspond to ionic strength of the solution,
and the concentration of fluorescein is small compared to ionic strength.
We used sodium as counterion. The dissociation coefficients of fluorescein yield the strongest influence of pH on mobility according to equations (6). The shaded zone corresponds to pH values where fluorescein
fluorescence is below 10% its maximum value.25,26 We used pK1 ¼ 4.45,
pK2 ¼ 6.8,26 and m2 ¼ 39.5 109 m2 V1 s1and m1 ¼ 25 109
m2 V1 s1,23,24 and a ¼ 4 Å. In the electrophoretic mobility calculations,
we neglected the effect of the cationic form of fluorescein (pK1 ¼ 2.1).
of electrolyte chemistry. For example, fluorescein mobility at pH
5 and 100 mM ionic strength is 11.6 109 m2 V1 s1; as
compared to a value of 37.3 109 m2 V1 s1 at pH 8 and 1
mM. We note that fluorescein’s quantum yield (which determines
fluorescence intensity) drops quickly for pH below about 5,25,26
and we indicate this in the plot with a shaded region.
Summary of ionic strength effects on pKa and fully ionized
electrophoretic mobility
In Part I, we summarized the effects of ionic strength on acid
dissociation constants, pK’s. In particular, we proposed a (rather
typical) Davies equation model (equation (40) of Part I) to
correct acid–base dissociation constants for ionic strength.
Above, we summarized how pK values strongly influence the
effective ion mobility of an arbitrary species. We saw how pK
values determine effective ion mobility by determining the
prevalence of ionic states and their associated fully-ionized
mobility (see eqs. (6) above). Further, we presented a correction
of the dilute approximation for fully-ionized electrophoretic
mobilities. Namely, we advised use of the Pitts equation to
approximate moX,z from mN
X,z (see equation (12)). We then
summarized how to determine effective mobilities for both corrected dissociation constants, pK’s, and corrected fully-ionized
mobilities, moX,z (see eqs. (4) and (12)). In Table 1, we summarize
these derivations and the various applications of concentrations
versus activities in both dilute and non-dilute solutions for
dissociation constants, mobilities, and conservation equations.
This journal is ª The Royal Society of Chemistry 2009
This journal is ª The Royal Society of Chemistry 2009
Lab Chip, 2009, 9, 2454–2469 | 2459
Concentrations
in terms of gX,z,
inifinite dilution
pKa, and
infinite
dilution
DcX;z
mobility mN
¼ V,ðmX;z cX;z EÞ
X,z
Concentrations28 Dt
þV:ðDX;z VcX;z Þ
þRX;z ðcY Þ
Infinite dilution mobility: mN
X,z
dcX
¼ kN cX cY
dt
11
Concentrations in fluxes,
activities in source terms
DcX;z
¼ V,ðmX;z cX;z EÞ
Dt
þV:ðDX;z VcX;z Þ
þRX;z ðgY cY Þ
Concentrations in terms of gX,z, finite dilution
pKa, and finite dilution mobility mN
X,z
Finite dilution mobility: mX.z
Activities
Davies equation for pKa, Pitts for mobilities. See also
equation (13).
4
moX;z ¼ mN
þ 0:16mN
X;z ð3:1 10
X;z Þ
pffiffiffi
I
pffiffiffi
1 þ 0:33a I
Davies equation for pKa, Pitts for mobilities. See
also equation (13).
Pitts equation12 for 1:1 electrolyte
Depends on pH electrode and
standard. Refer to Galster30
or Butler and Cogley31 for
discussion on pH scales.
See Levine11 chapter 23
NA
Davies (see Part I)
N
DpKX;1 ¼ pKX;1 pKX;1
pffiffiffi
pffiffiffi
¼ 1:02ð1 zÞ I =ð1 þ I Þ 0:3IÞ
Corrections (and example case)
a
Note that there are exceptions. For example enzymatic affinity constants are defined in terms of concentrations, but not derived from the law of mass action.32 b This does not only involve electrolytes
but also neutral species. The ideal dilute approximation is a hypothetical state where reactions are independent of ionic strength and diffusion limited.
Mass transport
Electrophoretic mobilities:
Weak electrolyte
Kinetic rate definition
(for the reaction
X + Y / XY)11
Electrophoretic mobilities:
Strong electrolyte
Concentrationsb11 Concentrations cX ¼ cX,0 + cX,1
(monovalent acid)
Activities pH h log10(gHcH)
(hydronium concentration)
Concentrations cX ¼ cX,0 + cX,1
(monovalent acid)
Concentrations pH ¼ log10cH (hydronium
concentration)
Conservation equations
pH
2
Activities KX,1 ¼ KN
X,1g ¼ cHcX,1/cX,0
Actual dependence (and example case)
Concentrations KN
X,1 ¼ cHcX,1/cX,0
Formulation for ideal, infinite
dilution case (and example case)
Chemical equilibriuma
(law of mass action)
(monovalent acid)
Equation
Table 1 Summary of example uses of activity and/or concentration in chemistry and mass transport equations. The 4th column gives examples of corrections when applicable
These principles and formulations are applicable to more general
formulations including generalized mass transport calculations.27
We note that diffusion coefficients also depend on ionic
strength. At finite ionic strength, the Nernst Einstein equation28
breaks down. For example, the effective diffusivity of potassium
chloride reportedly decreases from approximately 2.0 109 m2
s1 at infinite dilution to 1.8 109 m2 s1 at 100 mM, and
increases again at larger concentrations (e.g. 2.0 109 m2 s1 at
2 M).8 Harned and Nuttall29 showed good agreement between
their measurements and theoretical predictions of diffusion
coefficients by the Onsager and Fuoss theory. For detailed
treatment of diffusion coefficients in non-dilute solutions, refer
to Robinson and Stokes.8
Electrochemical effects
In this section we summarize electrolysis reactions that occur at
primary electrodes which drive current and flow in electrokinetic
systems. We describe a general model of potential loss applicable
to electrokinetic systems, and develop detailed models of electrochemical and Ohmic potential loses for common on-chip
capillary electrophoresis systems. We focus on the typical case of
direct current (DC) electrokinetic microsystems with aqueous
electrolytes wherein microchannel networks are connected to
end-channel reservoirs containing electrochemically stable electrodes (often platinum).
Reduction–oxidation reactions
Acid–base equilibrium is a subset of a more general concept of
reduction–oxidation (redox). A redox reaction involves the
transfer of n electrons, e, between species and follows the
general mechanism:
Ox + ne # Red,
(14)
where Ox is the oxidant (electron acceptor) and Red is the
reductant (electron donor). The forward reaction is termed
a reduction reaction, and the reverse an oxidation reaction. For
example, in acid–base equilibrium, the acid is an oxidant and its
conjugate base a reductant. The equilibrium between oxidant
and reductant is characterized by a redox potential, measuring
the affinity of the species to electrons. In the case of electrochemical cells, the redox potential is called the standard electrode
potential.
Redox reactions are at the basis of electrochemical
phenomena. The transfer of electrical current from a source to
a solution involves exchange of electrons at the interface between
the polarizable surface (the electrode) and the liquid. This
transfer generates the Faradaic current.
Electrolysis in electrokinetic systems
Consider a power source supplying a steady current to a microfluidic device, through two electrodes immersed in the electrolyte.
Redox reactions occurring at electrodes sustain current: an
oxidation reaction occurs at the anode, and a reduction at the
cathode. The net reaction is the sum of the ‘‘half reactions’’ of
oxidation and reduction, referred to as electrolysis. For EK
systems, water electrolysis is most often the dominant electrolytic
2460 | Lab Chip, 2009, 9, 2454–2469
Table 2 Electrode reactions for water electrolysis, the OER occurs at the
anode, and the HER at the cathode. The overall oxygen and hydrogen
evolution reactions depend on solution pH38,39
Acidic Solution
Anode
(OER)
Cathode
(HER)
Alkaline Solution
2H2O / 4H + 4e + O2
4OH / 4e + O2 + 2H2O
2H+ + 2e / H2
2H2O + 2e / H2 + 2OH
+
reaction.33–37 Water electrolysis is often referred to as the oxygen
evolution reaction (OER) at the anode, and the hydrogen
evolution reaction (HER) at the cathode.
The overall form of the OER and HER depends on the electrolyte pH, as summarized in Table 2. For the HER, the negative
surface charge of the cathode removes protons from solution.38,39
In acidic solutions, hydronium has relatively high concentration.
As a consequence, it is the dominant proton donor (i.e. electron
acceptor). In alkaline solutions, hydronium has lower concentration, and water becomes the dominant electron acceptor. In
the OER, the anode accepts electrons from hydroxide in alkaline
solution or from water in acidic solution. The effect is an increase
in pH at the cathode reservoir and a decrease at the anode
reservoir.33–37,40–44 Table 2 shows that supplying low-frequency**
or DC currents to electrokinetic systems result in several, often
undesirable phenomena. These include the generation or
consumption of hydronium or hydroxyde ions at the driving
electrodes, which result in spatial and temporal variations in
background electrolyte pH and conductivity.33–37,40–44 In addition, the generation of hydrogen and oxygen gas bubbles at
driving electrodes can complicate the ‘‘chip to world’’ connection
and can block or be entrained into a channel.37,41,45 Bubble
management and control of pH changes are key to optimization
and repeatability of EK experiments.
Electrolysis pH changes: Transport and mitigation strategies
As we mentioned above, in electrokinetic systems the anode
reactions tend to decrease local pH, while the cathode reactions
raise local pH. Transport of regions of perturbed pH from near
the electrode surface, into the reservoir volume and the channels
is a complex problem involving electrode reactions, multispecies
diffusion and electromigration, and fluid flow into and out of the
reservoir. Visualizations of the spatiotemporal development of
such pH fields was performed by Macka et al.36 Their setup
included 0.5 mm diameter platinum wires immersed into buffered and unbuffered chromate with xylenol blue dye as a pH
indicator, and acetate buffer with bromocresol green dye. Their
reservoirs were 2 mm deep (along the optical axis of their
imaging) and 10 by 40 mm in the image plane. Their visualizations show that, for low viscosity electrolytes, convective fluid
transport greatly contributes to the spatiotemporal development
** A simplified model of the electrode/electrolyte interface is that of
a capacitor and resistor in parallel, where the capacitor models electric
double layer charging and resistor charge transfer across the
interface.47 At high enough frequencies, the current supplied to the
electrode will serve mostly to charge the double layer, and negligible
charge is allowed to cross the interface (negligible Faradaic current)
before polarity reverses.47
This journal is ª The Royal Society of Chemistry 2009
of the pH field in the reservoir. In any case, the general problem is
quite complex and can involve thermal convection, bubble
nucleation and growth phenomena, bubble detachment and
bubble-driven flow, and coupled convective-diffusion of pH
fields.
There are several strategies for mitigation of electrokinetic
buffer depletion. Theses include appropriate use of buffers to
ensure pH stability (cf. Part I); frequent buffer replenishments,33,34,42 the use of low conductivity (e.g., zwitterionic)
buffers to reduce current,37,40,44 large electrode-to-channel
distances to confine reaction zones;36,37 and increasing reservoir
size37 to both confine reaction zones and improve total buffering
capacity. However, one clear advantage of on-chip CE type
applications over traditional systems is their ability to deal with
small sample volumes.37 Therefore increasing reservoir size and
electrode-to-channel distances is often counterproductive.
Predictive models for transient reservoir pH have been presented
by Bello33 and Corstjens et al.,34 and these provide guidelines for
the required frequency of buffer replenishment. Below, we
present a model allowing estimation of pH changes in a typical
microfluidic EK experiment.
pH changes during an electrokinetic experiment weak electrolyte
buffer
Consumption and production of hydronium and hydroxide ions
in the reservoirs (cf. Table 2) can affect electrolyte chemistry and
impact the performance of the entire system. We here model and
discuss quantitatively the effects of concentration changes
induced by electrode reactions.
We present an illustrative model which accounts for the effect
of electrode reactions on a system composed of a single channel
connecting two reservoirs as shown in Fig. 2. The buffer system is
a weak base B (e.g., Tris base), titrated with a strong acid HA
(e.g., hydrochloric acid HCl) to a desired pH. We will later
present results of the analogous problem of a weak acid titrated
with a strong base. We perform a control volume analysis with
the following assumptions: (i) electrode reactions are solely water
electrolysis, (ii) electroneutrality applies everywhere in the
system, (iii) hydronium and hydroxide ions do not carry current,
(iv) the solution is dilute (so that moX,z ¼ mN
X,z and pK ¼ pKI ¼ 0),
(v) advective current is negligible (in particular electroosmotic
flow is negligible), and (vi) the reservoirs are ‘‘well stirred’’ so the
concentrations are uniform within each reservoir. Although
simplified, this example problem is sufficient to summarize and
review relevant dynamics of these problems and point out key
parameters. As discussed and visualized by Macka et al. pH
changes originate at the electrode surface and are transported by
diffusion and advection (e.g., due to thermal convection and bulk
flow from the reservoir into the channel). Assuming negligible
thermal convection (i.e., sufficiently small reservoirs), we can
assume pH perturbations originate at the electrode and transport
outward. In such cases, we hypothesize that a well-stirred
reservoir approximation will overpredict the effects of electrode
reactions on the channel flow. We therefore take this well-stirred
assumption as a conservative estimate in estimating experiment
durations and reservoir volumes required to avoid pH changes in
a connecting microchannel(s).
We consider a constant voltage experiment, but the model can
be readily modified for (the simpler case of) constant current
conditions. The potential drop between anode and cathode Vapp
generates an electric field of magnitude Vapp/L in the channel.
The current i sets the rate of creation of hydronium ions at the
anode and hydroxide ions at the cathode. As in most experimental studies, we assume constant voltage, and so the current
adjusts to the electrolyte conductivity in the channel, sch, where
the superscript ch designates the channel. For a channel of cross
sectional area S, the current is simply i ¼ Vapp(schS/L). Applying
ch
electroneutrality inside the channel (cch
A,1 ¼ cB,+1 under
7
‘‘moderate pH’’ conditions ), and under the assumption that only
buffer species carry current (i.e., ‘‘safe pH’’), the conductivity
N
N
N
simplifies to sch ¼ Fcch
A,1(mB,+1 mA,1). Recall that mX,z is the
fully ionized mobility of X in valence state z.
The Faradaic current induces migration of the anion A from
the cathode to the anode. The fluxes of A into the anode
reservoir and out of the cathode reservoir are respectively JaA ¼
ch
c
N
c
mN
A,1cA,1(VappS/L) and JA ¼ mA,1cA,1(VappS/L), where the
superscripts a and c designate respectively anode and cathode
reservoirs. Current flow therefore depletes the cathode reservoir
of buffer anions (replacing them with hydroxyde ions). We can
write the conservation of A in the cathode reservoir:
N
dccA;1
mA;1 Vapp S
;
¼ ccA;1
YL
dt
Fig. 2 Schematic of an electrokinetic experiment where the electrolyte is
a solution of a weak base titrated with a strong acid. The anion A
migrates towards the anode. The cation BH + migrates towards the
cathode with an effective mobility determined by its ionization state (and
local pH). The anode and cathode respectively lower and raise local pH
over time.
This journal is ª The Royal Society of Chemistry 2009
(15)
where Y is the reservoir volume (equal for anode and cathode).
This first order ordinary differential equation yields an exponential decay of the concentration of A from its initial value c0A,1
N
with characteristic time scale s ¼ YL/(mN
A,1VappS) (mA,1 < 0 so
that s > 0) in the cathode reservoir:
ccA,1(t) ¼ c0A,1 exp(t/s).
(16)
Similarly, the rate of change of anion concentration at the
anode is
Lab Chip, 2009, 9, 2454–2469 | 2461
N
dcaA;1
mA;1 Vapp S
:
¼ cch
A;1
YL
dt
(17)
The conservation of A in the channel JcA + JaA ¼ 0 yields
ccA,1 ¼ cch
A,1, so that (17) becomes:
dcaA;1 c0A;1
¼
expðt=sÞ;
dt
s
We
note
that
from
the
initial
equilibrium
0
c0A,1 ¼ c0B(1 + 10pH pKa)1 where the buffer is initially titrated
to pH0. We then find the pH in the anode reservoir from the
Henderson-Hasselbalch equation (see eq. (8) of Part I):7
pHa ¼ pKa þ log10
(18)
¼ pKa þ log10
which after integration yields:
caB caA;1
caA;1
#
#
h10pH0 pKa þ f ðf 1Þ1 f 1 þ 2t=s þ expðt=sÞ
1 :
2 expðt=sÞ
(25)
(19)
(20)
N
N
where f ¼ mN
B,+1/(mB,+1 mA,1) is the transport number,
measuring the contribution of the cation to current conduction.
Similarly, the flux of B out of the anode is:
a
JaB ¼ maBcaB(VappS/L) ¼ mN
B,+1cB,+1(VappS/L).
(21)
Note caB and maB are respectively the total concentration and the
effective mobility of species B at the anode,7 and these describe
total outflow of B and BH+. caB,+1 and mN
B,+1 are the respective
concentration and mobility of (specifically) the cation state
BH+. Applying electroneutrality to the anode reservoir, we have
a
JaB ¼ mN
B,+1cA,1(VappS/L). In the anode, electromigration
depletes the buffer of species B, replacing it with protons. In the
same manner as the case of the anode and equation (15), we can
derive the rate of change of weak base in the anode reservoir:
dcaB JBa caA;1 mN
B;þ1
¼
¼
:
dt
Y
s mN
A;1
We recognize that mB,+1/mA,1 ¼ f/(f 1), so that:
t
caB ¼ c0B þ c0A;1 f ðf 1Þ1 2 þ expðt=sÞ 1 :
s
N
(26)
We note from equations (25) and (26) that the variations of pH
at the cathode or the anode do not depend on initial buffer
concentration. That is, at constant applied voltage, the pH
changes due to electrode reactions are independent of buffer
concentration. This result is intuitive since both buffer capacity
Table 4 Summary of pH changes in the anode and cathode for a weak
base buffer strong acid titrant, and a weak acid buffer, strong base titrant
equation
in the limits where t s and pH is initially close from pKa. Each
0
is an expression of DpH ¼ pH pH0. The parameter a ¼ 10pH pKa is an
indicator of buffer strength
(22)
N
(23)
Anode
Weak base buffer,
strong acid titrant
Weak acid buffer,
strong base titrant
að1 f Þ t
s
f
Similarly, at the cathode well:
ccB ¼ c0B + c0A,1f(1 f)1(1 exp(t/s)).
Cathode
f ða 1Þ þ a þ 1 t
s
að1 f Þ
.
FYc0A;1
ð1 f Þ1 expðt=sÞ;
s
a þ 1 af t
að1 f Þ s
aþf t
s
f
.
i¼
We can derive the pH variation at the cathode in the same
manner:
#
c
cB ccA;1
c
pH ¼ pKa þ log10
ccA;1
h
i
0
¼ pKa þ log10 10pH pKa þ ð1 f Þ1 expðt=sÞ ð1 f Þ1 :
.
Using the expression of conductivity, we can derive the
expression of current as a function of time:
.
caA,1(t) ¼ c0A,1[2 exp(t/s)].
(24)
Table 3 Summary of pH changes in the anode and cathode for a weak base buffer titrated with a strong acid, and a weak acid buffer titrated with
a strong base. Each equation is an expression of pH pKa
Anode
Weak base
buffer,
strong acid
titrant
Weak acid
buffer,
strong base
titrant
log10
Cathode
8
0
<10pH pKa þ f ð f 1Þ1 f 1 þ 2t=s þ expðt=sÞ
:
log10{10pH
2 expðt=sÞ
0
+ pK
1
9
=
log10{(10pH
pKa
+ (1 f)1)exp(t/s) (1 f)1}
;
(
+ (f1 1)[exp(t/s) 1]}
2462 | Lab Chip, 2009, 9, 2454–2469
0
log10
)
0
10-pH þpKa þ f 1 f 1 1 2t=s þ expðt=sÞ
1
2 expðt=sÞ
This journal is ª The Royal Society of Chemistry 2009
Fig. 3 Effect of electrode reaction on anode pH and cathode pH for Tris and acetate buffers titrated with respectively hydrochloric acid and sodium
hydroxide. (a) and (b): pH variation for 10 mM Tris initially titrated with hydrochloric acid to pH0, which is varied from pKa 1 to pKa + 1. We consider
2.5 kV applied to a 5 cm long, 50 mm wide (20 mm deep) channel. (a) pH at the 100 ml anode reservoir. If initial pH is below pKa, the anode pH drops
dramatically, as the anode generates H+. (b) pH at the 100 mL cathode reservoir. The pH change is slow as the cathode reservoir fills with weak base.
Inset: anode (black solid lines) and cathode (grey solid lines) pH for 10 mL, 25 mL and 100 mL reservoirs and pH0 ¼ pKa. (c) and (d): pH variation for 10
mM acetate buffer (titrated with sodium hydroxide) in otherwise the same conditions as the previous case. (c) pH at the 100 mL anode reservoir. The pH
change is slow as the anode reservoir fills with weak acid. Inset: anode (black solid lines) and cathode (grey solid lines) pH for 10 mL, 25 mL and 100 mL
reservoirs and pH0 ¼ pKa. (d) pH at the 100 mL cathode reservoir. If initial pH is above pKa, the cathode pH increase dramatically, as the cathode
generates OH.
and current are directly proportional to buffer concentration.
The two important parameters are: (i) the time scale
s ¼ YL/(mN
A,1VappS); and (ii) the initial buffer pH. We
summarize the closed form solutions for pH at the anode and
cathode of a weak base buffer system (derived above) and a weak
acid buffer system (derivation not shown) in Table 3.
For convenience and quick estimates, we summarize in Table 4
the same relations but in the limit where the experiment time is
much smaller than the characteristic time scale, s. s ranges
between 1 and 100 min for typical EK systems (see examples
below). The parameters in Table 4 can be interpreted as a fairly
general set of non-dimensional parameters describing the
importance of pH changes due to electrolysis in buffer reservoirs.
These expressions are simplified and yet capture the effects of
0
proper titration (via the parameter a ¼ 10pH pKa), the effect of
ion mobilities (via f), and the relative magnitude of the experiment time to the characteristic time constant. The latter depends
on buffer concentration, reservoir volume, and applied voltage.
Absolute values of the parameter t/s smaller than 0.05 are
a good indication that pH changes should be negligible.
Fig. 3 shows examples of pH changes in 100 ml (each) anode
and cathode reservoirs during an electrophoresis type experiment
for two buffer systems: 10 mM Tris (pKa ¼ 8.2) titrated with HCl
or 10 mM acetic acid (pKa ¼ 4.8) titrated with NaOH. pH drops
dramatically in the anode reservoir for the Tris buffer system for
the cases where initial pH is below its pKa. Conversely, we see pH
This journal is ª The Royal Society of Chemistry 2009
increases dramatically in the cathode reservoir for the acetate
buffer system for cases where the initial pH is higher than its pKa.
More generally, pH changes dramatically in reservoirs towards
which the titrant ion migrates; and pH changes relatively slowly
in the other reservoir. Further, we see that rapid variations in the
anode and cathode can, in all cases, be limited by titrating the
buffer to respectively higher and lower pH. For example, Fig. 3a
shows that if Tris is titrated slightly above its pKa, it buffers more
efficiently at the anode (while acetate buffer titrated with slightly
below its pKa buffers more efficiently at the cathode, as in
Fig. 3d). In this way, anode and cathode reservoir chemistry can
be slightly adjusted to ‘‘anticipate’’ the effects of the anode and
cathode reactions.†† The insets show the expected result that
larger reservoirs are more robust to pH changes in both acid and
basic buffer systems. Lastly, we note that pH changes are independent of buffer strength only at constant voltage, as it is only
dependent on the amount of charges (Coulombs) transferred to
the liquid (and both charge transfer and current in and out of the
reservoirs is proportional to buffer strength). We note, however,
†† One strategy in systems with significant electroosmotic (or other bulk)
flow may therefore be to slightly adjust buffer pH to ‘‘anticipate’’ the
change in the flow inlet reservoir. For example, for a Tris
hydrochloride buffer in glass channels and pure electroosmotic flow
(no pressure-driven flow), you might titrate the buffer to favor the
anticipated creation of acid in the anode reservoir. Accordingly, you
might titrate to 0.5 pH units above the Tris pKa.
Lab Chip, 2009, 9, 2454–2469 | 2463
that buffering strength is a critical parameter in constant current
experiments.34
Back-of-the-envelope calculation for importance of pH change
due to electrolysis
In most electrokinetic applications, we are dealing with moderate
pH conditions and also the conductance between two endchannel electrodes is approximately constant. For such cases, the
time scale for pH change due electrolysis current i in a reservoir
volume Y can be expressed as s ¼ Ys/(izXmN
X ) (since i ¼ s(Vapp/L)S)
for both the anode and cathode reservoirs. Here, mN
X is the
mobility of the titrant (i.e., strong acid or base). This time scale
can be used to provide a rough, order-of-magnitude estimate for
the time before we should expect a significant change in pH.
For example, consider typical values of Y ¼ 5 ml, i ¼ 20 mA, and
s ¼ 0.5 mS/cm (consistent with 10 mM Tris titrated with HCl to
pH ¼ 8.2, 500 V cm1 field in a D-shaped channel 50 mm wide,
15 mm deep channel). For a typical buffer such as Tris titrated
N
¼ 79.1 109 m2 V1 s1. This yields
with HCl, mN
X ¼ mCl
s x 3 min. Provided the electrode is placed in the reservoir away
from the channel entrance, experiments with durations significantly shorter than this (less than 20 s or shorter) should not
suffer from significant pH change. This example is easily scalable
to other systems; so increasing the reservoir to 50 ml yields s ¼ 30
min. The latter suggests steady pH operation for order 3 min.
Our results are in qualitative agreement with the work of
Corstjens et al.34 who studied the effect of electrolysis in constant
current CE experiments at finite dilution. Bello33 studied a more
complex buffer system and modeled electrolysis effects in
constant voltage CE experiments with no assumption on pH
(i.e., hydronium and hydroxide can have a significant concentration). The parameters from both studies are perhaps more
suitable to classical CE systems (i.e., channels longer than 10 cm,
and reservoir volumes larger than 0.5 mL) but are qualitatively
similar to a microfluidic electrokinetic case.
We derived a simple expression for the pH changes in wellstirred reservoirs. We note that more general treatments of this
problem can be very complex. For example, the physics involved
in the ionic transport within the reservoirs is in general
a complex, multi-species reaction equilibrium transport problem.
Possible additional phenomena include thermally driven buoyancy fluid motion near the electrode;36 surface tension effects due
to the change of contact angle between fluid surface and electrode;46 flow induced by bubble nucleation and growth; flow
induced by bubble detachment and bubble motion toward the
top of the reservoir; flow in reservoir induced by electroosmotic
flows; and pressure gradients induced by changes in reservoir
volume. The visualizations of Macka et al.36 confirm that the
spatio-temporal distribution of pH in an electrophoresis system
reservoir is indeed a complex process.
Electrolysis bubbles
Water electrolysis generates oxygen gas at the anode and
hydrogen gas at the cathode (see Table 2). More precisely,
hydrogen and oxygen atoms adsorbed onto the cathode and
anode electrode surface respectively, react to form gaseous O2 or
H2.47 Electrolytic gas bubbles form at nucleation sites on the
2464 | Lab Chip, 2009, 9, 2454–2469
electrode surface or supersaturation regions in the electrolyte
near the electrode interface.48 The growth of adsorbed bubbles
shields the electrode surface, decreasing the electrode real surface
area, Sr, and dropping the system limiting current.48,49 Bubbles
may eventually separate from the electrode surface and leave the
reservoir driven by buoyancy or may in some cases be advected
into the channel, causing blockage.37,41 Adsorbed or detached
bubbles in the channel or reservoir increase interelectrode resistance,49 resulting in a current drop for potentiostatic operation.
Note that, for the same current, the rate of hydrogen production
at the cathode is twice that one of oxygen at the anode.
Several mitigation strategies applicable to on-chip capillary
electrophoresis can prevent bubble entrainment into the channel
or bubble generation entirely. Displacing electrodes from the
channel entrance may be effective in preventing bubbles from
entering the channel.37 Several groups have used ion exchange
membranes such as Nafion to prevent bubble advection into the
channel.50,51 Moini et al.52 demonstrated that introducing
hydroquinone (HQ) into the buffer allows for HQ oxidation to
replace water oxidation at the anode, resulting in the formation
water soluble p-benzonquinone. Recently, Kohlheyer et al.53
used quinhydrone to prevent both oxygen and hydrogen bubble
formation, which at their experimental conditions was effective
up to approximately 40 mA. Another possible method is operation in a degassed electrolyte which allows dissolution of gases
produced at the electrode.54 Lastly, palladium electrodes have
been shown to absorb some hydrogen, and thus may in some
cases help mitigate hydrogen gas formation at the cathode.55
Potential losses in CE systems
Consider a simple electrokinetic system consisting of a microchannel connecting two reservoirs each containing an electrode.
The potential applied to the electrodes of this system does not
entirely drop across the microchannel. Instead, finite potential
differences exist between the electrode surface and channel
entrance. This is due to (i) a potential difference across the
electrode-electrolyte interface which drives electrochemical
reactions occurring at electrode surfaces, and (ii) a potential drop
between the electrode surface and channel entrance associated
with current through the reservoirs (an Ohmic loss). Many onchip electrophoresis systems use high voltages (100 V to 10 kV)
and so are largely insensitive to the typically small voltage losses
associated with electrochemical reactions (as will be shown in
Fig. 6). However, these losses can become important in systems
which use relatively low voltage (e.g., smaller than 20 V).
Examples of these include low voltage electrophoretic separations,56–59 and low voltage electroosmotic pumping.60,61 In
contrast, Ohmic loss in the reservoirs can form a significant
portion of applied potential for both low and high voltage
systems (as will be shown in Fig. 5). Accounting for reservoir
potential drop is particularly important for analyzing strategies
which involve changing reservoir geometry or electrode placement to confine electrode reactions.36
The voltage applied to a typical single channel device can be
expressed in terms of the potential drop across the channel, Vc,
and several sources of potential loss as follows:
Vapp ¼ Vc + Eeq + ha + hc + Vres,ar + Vres,cr.
(27)
This journal is ª The Royal Society of Chemistry 2009
Eeq is the equilibrium potential of redox reactions, and can be
computed from the Nernst equation.62 The Nernst equation
relates the standard electrode potential with the effects of nonstandard pressure, temperature, and solute activities. Bard62
(Appendix C) and Bockris47 (Chapter 7, p. 1352) list standard
electrode potentials in aqueous solutions for a variety of reduction reactions. A Faradaic current, or the driving current
required for typical electrokinetic experiments, is theoretically
possible only for Vapp > Eeq. The overpotential, h, represents the
potential required to drive the rate limiting step of the electrode
reaction at the desired Faradaic current.47 Vres is the Ohmic
potential drop maintained between the electrode surface and the
channel entrance. The second subscripts in equation (27) refer to
the anode (a) cathode (c), anode reservoir (ar), and cathode
reservoir (cr), respectively.
We leverage the Tafel equation for overpotential and Ohm’s
law for Vc and Vres to express potential drops in our system as
follows:
!
ðL
I
I
dx þ Eeq þ ba ln
þ.
Vapp ðtÞ ¼
sðx; tÞS
jo;a Sr;a
(28)
0
I Leff
I Leff
þ
:
sar ðtÞ Seff scr ðtÞ Seff
surface area, meaning the surface area on which electron
transfer occurs. For platinum wire electrodes, Sr differs from
the simple wire geometric surface area, Sgeom, due to surface
roughness.62,66 Real surface area of a platinum wire electrode
can be estimated by a variety of in-situ or ex-situ experimental methods, which are extensively described and evaluated by Trasatti et al.66
We invoke the Tafel equation by assuming high overpotential
(greater than approximately 100 mV), and that concentration of
OER and HER reactants at their respective electrode surfaces is
within 10% of the reactant concentration in the bulk electrolyte.62
The high overpotential assumption is reasonable for the OER
reaction at currents characteristic of electrokinetic experiments
(order mA currents and above), due to the low OER exchange
current density (see Fig. 6). For simplicity, we additionally
assume jo,a and ba are not functions of applied potential.47
Despite further complexities associated with precise modeling of
electrode reactions (e.g., including the effect of pH changes in
reservoirs on HER and OER overpotentials39), the model presented here allows for basic parametric studies and insights into
the sources of these potential loses.
sar, scr, and s are respectively the conductivity in the anode
reservoir, cathode reservoir and channel. Our major assumptions
for the expression of Vc include 1-D conductivity gradients in the
microchannel, and negligible contribution of bulk fluid motion
to ionic current (i.e., negligible advective current).47 The latter
assumption allows for modeling Vc using Ohm’s law. In
modeling Vres, we have assumed spatially uniform conductivity
within each reservoir and reservoirs with identical geometry.
Note that reservoir conductivity may change in time due to
addition or removal of charged species by the electrochemical
reactions summarized in Table 2. S is the channel cross sectional
area, and Seff and Leff are effective parameters which represent
the cross sectional area and path length of the ionic current in the
reservoir. These parameters account for three-dimensional
reservoir geometry and current distribution, and will be discussed further below.
For the Tafel equation, the parameter b is a function of
temperature and a symmetry factor, and dividing b by the
factor 2.3 yields a parameter known as the Tafel slope.47 Tafel
slope has been reported as roughly 120 mV/dec (i.e. mV per
‘‘decade’’ slope in a semilog plot) for the HER and
OER.‡‡47,63,64 The exchange current density, jo, physically
represents the bi-directional current at equilibrium (at Vapp #
Eeq), and is often used to describe the kinetics of the chemical
reaction at the electrode.47 With platinum electrodes, the
exchange current density of the HER and OER are of the
order 104 A cm2 and 109 A cm2 respectively, for both
acidic and basic conditions.65 As a result, for water electrolysis the anode exchange current density, jo,a, is several orders
of magnitude smaller than jo,c, and thus ha [ hc. We have
thus neglected hc in equation (28). Sr is the real electrode
We here estimate the relative importance of the Ohmic potential
drop in electrokinetic reservoirs for typical microchannel
geometries. We consider a cylindrical reservoir with a diameterto-liquid-height ratio of unity and an approximately cylindrical
microchannel connected to the reservoir at the bottom right, as in
Fig. 4. The electrode is a thin, vertical, and cylindrical wire placed
along the wall of the reservoir diametrically opposed to the
channel as shown. We recommend the latter placement in
experiments as: (i). the wire is kept well away form the channel
entrance to mitigate the effects of pH changes and bubbles
generated by the electrode; and (ii). placing the tip of the wire at
the bottom of the reservoir is easier to reproduce (vs. suspending
the wire part way down the reservoir). Assuming approximately
uniform conductivity within each reservoir, the geometric
parameter Leff/Seff can be calculated by use of a numerical
solution to the conservation of current relation assuming
uniform (locally, for each domain) conductivity, which yields:
‡‡ We note that reported Tafel slope varies between about 50 and 200
mV/dec for a range of electrode materials, electrolyte chemistry, and
pH.47,63,64 We report here 120 mV/dec only as a typical value.
This journal is ª The Royal Society of Chemistry 2009
Estimates of Ohmic voltage drop in typical reservoirs
V2f ¼ 0,
(29)
where f is local potential. As shown in Fig. 4, we modeled
potential drop in a reservoir connected to a 0.5 mm long
microchannel section. The channel length is significantly longer
than the length yielding approximately uniform and parallel
electric field in the channel (see Fig. 4). We rounded the edges
of the transition between reservoir and channel using fillets with
10 mm radii of curvature. The boundary conditions are: specified
potential at the electrode surface, Velec; specified potential
surface 0.5 mm into the small channel; and zero normal current
at all other surfaces. The three-dimensional potential field was
solved with a commercially available finite element simulation
software (Comsol Multiphysics 3.3a, Burlington, USA). Solutions were obtained using over 8 104 tetrahedral mesh
elements; were mesh size independent; and converged with over
8 orders of magnitude decrease in the residual. To estimate the
Lab Chip, 2009, 9, 2454–2469 | 2465
To achieve this, we explored six geometries described by the
following parameters: channel diameters d of 20 and 100 mm;
reservoir diameters D of 2, 6, and 10 mm, reservoir diameter-toheight ratios of unity, and a wire electrode of 0.2 mm diameter.
We found that Leff/Seff is insensitive to reservoir and channel
diameter. For these conditions, the value of Vres is dominated by
potential loss occuring near the channel entrance. Fig. 4 shows
crowding of equipotential and current lines near the channel
entrance. Simulations also show that Leff/Seff increases by
a factor of approximately 2.5 in changing channel diameter from
100 to 20 mm, indicating a higher Ohmic resistance for the
reservoir when connected to the smaller channel. These results
suggest that, for this range of dimensions, reservoir potential loss
is a strong function of channel diameter but not of reservoir
height and diameter.
To further explore the impact of Vres in electrokinetic systems,
we define a parameter, a1, describing the ratio of the Ohmic
resistance of the reservoirs to the total system resistance,
assuming approximately negligible overpotential and equilibrium potential. First, for the case of uniform conductivity, we
have:
2Leff =Seff
a1 z
:
(32)
2Leff =Seff þ L=S
Fig. 4 Model geometry and results showing evenly-spaced equipotential
lines and dashed current lines in the x-y midplane of a D ¼ 6 mm (height
and diameter) reservoir. Detail view shows the near-channel region of the
reservoir, and current lines are shown as they continue into the d ¼ 100
mm channel. The potential drop in the reservoir is dominated by the drop
in the near-channel region (where stream lines crowd). This effect
explains insensitivity of ionic resistance to reservoir size, and allows us to
conclude that for this typical configuration (i.e., vertical wire touching
bottom of well) and geometries (D ¼ 2–10 mm, d ¼ 20–100 mm), potential
loss between the electrode surface and channel entrance is largely independent of D. Instead, reservoir potential loss is highly sensitive to
microchannel diameter, d. Also note that current lines, and thus electric
field lines, become parallel after approximately a distance d into the
channel.
geometric parameter Leff/Seff associated with the reservoir, we
first compute the area-average potential at the channel inlet,
Ventr, from model results; and then define the potential loss in this
reservoir, Vres, as follows:
Vres h Velec Ventr.
(30)
We then calculate the system current, i, from model results,
and relate Vres and i to the geometric parameter as follows:
Leff Vres sM
;
¼
Seff
i
(31)
where the value of the geometric parameter is independent of the
conductivity assigned in our model, sM. We performed a parametric study to investigate the effect of changing reservoir
characteristic dimension D, on the geometric parameter Leff/Aeff.
2466 | Lab Chip, 2009, 9, 2454–2469
Next, we define a similar parameter, a2, describing the ratio of
Ohmic resistance of the reservoirs to the total system resistance in
the case where the buffer in the anode reservoir has a different
conductivity than that in the channel and cathode reservoir:
Leff =Seff ð1 þ gÞ
;
(33)
a2 z Leff =Seff ð1 þ gÞ þ ðL=SÞ
Fig. 5 The ratio of reservoir to total system resistance, a1, as a function
of channel length, L, and channel diameter, d, for a two-reservoir, one
channel device with uniform buffer conductivity. The curves in this plot
and inset apply to reservoir characteristic dimensions, D, from 2 to 10
mm, as the Leff/Seff geometric parameter is unchanged for these D.
Reservoir potential loss remains below 1% of Vapp (dotted grey line) for
20 mm diameter channels, but rises above 1% when L < 1 cm for 100 mm
channels. Reservoir losses contribute to a higher fraction of Vapp for wider
microchannels. The inset shows ratio of reservoir to total system resistance, a2, for the case of uniform cathode reservoir and microchannel
electrolyte conductivity, but relatively low anode reservoir conductivity.
We plot a2 versus microchannel length and conductivity ratio, g, for the
case of d ¼ 100 mm. The ratio of reservoir to system resistance, a2, increases
with conductivity ratio, g. For a 100 mm diameter channel, reservoir
potential drop is approximately 80% of Vapp at g ¼ 103, L ¼ 1 cm.
This journal is ª The Royal Society of Chemistry 2009
where g is the relevant conductivity ratio, scr/sar. The latter case
is relevant to systems which employ non-uniform conductivities
such as field amplified sample stacking or isotachophoresis.57,67
In Fig. 5, we show the parameter a1 as a function of microchannel length. For standard electrophoresis chips with uniform
conductivity throughout the system and typical geometries (L 1–5 cm, D 2–10 mm, d < 100 mm) the potential drop in both
reservoirs remains below 1% of Vapp (a1 < 0.01). In typical
uniform conductivity systems, reservoir Ohmic potentials are
nearly always negligible.
The case of non-uniform system conductivity is very different.
Here, reservoir potential loss can be a large percentage of applied
potential. In the inset of Fig. 5 we plot a2 versus channel length,
but now for various channel-to-reservoir conductivity ratios, g.
For a 100 mm diameter channel, with L ¼ 1 cm, and a conductivity ratio of 103, the potential drop in the anode reservoir
approaches 80% of Vapp.
Note that the results shown in the inset of Fig. 5 represent
a quasi-steady, ideal condition wherein conductivities in each
region (channel and reservoir) are well described and uniform. In
practice, the problem can be unsteady and strongly dependent on
the direction of bulk flow in the system. For example, consider
the case where the channel and anode reservoir are filled with
high conductivity solution; and the cathode filled with low
conductivity solution. If bulk fluid flows from anode to cathode
(as in typical electroosmotic flow systems with negative zeta
potentials28), then high conductivity liquid quickly enters the
cathode reservoir from the channel. This should quickly modify
the high resistance region near the channel-to-reservoir interface
(cf. detail view in Fig. 4), quickly lowering the cathode reservoir
Ohmic resistance. Such a case should be very different from, say,
anode-to-cathode bulk flow with low conductivity only in the
anode reservoir. In the latter configuration, a large Ohmic
potential drop in the anode reservoir should persist, since it
remains filled with only low conductivity.
General effect of overpotential and equilibrium potential on EK
systems
We here investigate the effect of overpotential, ha, and equilibrium potential, Eeq, on typical DC electrokinetic systems.
Generally, these potential loses are on the order of 1 V, and can
therefore be neglected for all but low voltage electrokinetic
systems. As shown in Fig. 6, at low applied voltage, Vapp is
dominated by overpotential and equilibrium potential. This
regime is called the activation region (with reference to the
overpotential and equilibrium potential being the necessary
voltage to ‘‘activate’’ the electrode reactions and reach the
required Faradaic current). At high applied voltage, Ohmic
potential dominates in what we term the Ohmic regime. The real
area of the electrode surface, Sr, is an important parameter for
low potential applications, but has negligible effect on Vapp at
high potentials. This is shown in the inset of Fig. 6 where we plot
overpotential versus applied current for a range of real areas. For
low DC voltage electrokinetic applications, minimizing power
loses can include strategies such as: maximization of Sr, reduction of Eeq by use of electrodes or electrolytes which result in
lower equilibrium potentials,59 and maximization of jo,a by use of
the best available electrocatalyst as electrode material.47
This journal is ª The Royal Society of Chemistry 2009
Fig. 6 A plot of Vapp and its constituents versus current at representative
parameters for on-chip electrokinetic devices: Eeq ¼ 1.23 V,62 s ¼ sa ¼ sc
¼ 0.075 S m1, L ¼ 5 cm, d ¼ 100 mm, D ¼ 2–10 mm, Leff/Seff ¼ 6380 m1
(from finite element model solution described previously), and Sr ¼
10Sgeom for a 6 mm high and 0.2 mm diameter wire electrode. We used the
exchange current density and Tafel slope for the OER on platinum
electrodes.47,63–65 We observe a transition between activation and Ohmic
regimes at approximately 10 nA. Operating well into the Ohmic regime,
i.e. at currents above 1 mA, allows for neglecting overpotential and
equilibrium potential losses as they represent a negligible fraction of Vapp.
The inset shows that over eight orders of magnitude of Sr, the variation of
ha is order one volt. Therefore Sr must be precisely determined to model
low voltage EK devices, but not for typical high voltage experiments.
Electrode materials for on chip EK systems
We here focus on electrode materials applicable to on-chip
electrokinetic devices, particularly for electrodes introduced at
end-channel reservoirs which supply DC current. For these
systems, platinum electrodes are most common due to the
electrochemical stability and catalytic ability.47 For typical
microfluidic applications, a small length of platinum wire
(approximately 1 cm) is connected to the ends of standard high
voltage leads (e.g., copper wire). Note that platinum theoretically dissolves at potentials above 1.2 V.47 Consequently, platinum cannot be considered entirely electrochemically stable.
Gencoglu et al.4 performed in situ cyclic voltammetry measurements and ex situ SEM imaging to visualize platinum oxidation
and dissolution at wire electrodes driving electric fields up to
approximately 300 V.cm1. They observed both chemical and
morphological changes to the platinum wire surface in several
electrolytes, including 50 mM Tris-buffered saline and 20 mM
HEPES-buffered saline. A major drawback in the use of platinum, especially towards development of commercially available
electrokinetic devices, is high cost. Baldwin et al.68 developed
sputtered platinum driving electrodes, using platinum layers that
are several hundred nanometers thick. Low-cost printed
graphite ink electrodes have been successfully implemented as
primary driving electrodes in microchip CE devices.69 However,
some carbon-based electrodes have electroactive surfaces and
may not be suitable as robust primary electrodes.70 An interesting set of less expensive materials which merit further investigation for electrokinetic systems are dimensionally stable
anodes (DSA), which combine an unstable catalyst and a stabilizing oxide in a thin film supported by a conducting metallic
substrate.71,72
Lab Chip, 2009, 9, 2454–2469 | 2467
Conclusion
Acknowledgements
We have reviewed of the coupling between acid–base equilibria,
electrophoresis, and electrochemistry in typical DC electrokinetic devices. We first presented a brief summary of ion transport
by electrophoresis. In particular, we discussed the relation
between electrophoretic mobility and species dissociation reactions. We derived a general formulation for the effective electrophoretic mobility of a weak electrolyte with an arbitrary
number of dissociations. We also discussed the departure of
mobility data from infinite dilution approximations. The latter
included presentation of a Pitts model for ionic strength
correction of the ideal, infinite dilution mobility. As an illustrative example, we showed the variation of predicted effective
electrophoretic mobility (based on known pKa and fully-ionized
mobility data) for fluorescein as a function of pH and ionic
strength.
We presented an analysis of electrode reactions on buffer
systems in typical microfluidic electrokinetic experiments. We
derived a closed form expression for the temporal variation of
pH in well-stirred reservoirs for buffers composed of a weak
electrolyte and a strong titrant. We showed that pH can
significantly change if the electrolyte is not well buffered and/or
the reservoirs are not large enough for the current and time
frame of interest. We showed that the case of the anode and
cathode are very different and depend on whether a weak acid
or a weak base are used for buffering. We also derived a set of
non-dimensional parameters useful in estimating pH effects in
reservoirs.
We then described the effects of electrode reactions on
microfluidic electrokinetic systems. We first discussed electrolysis. The nature of electrode reactions depends on electrode
polarity and electrolyte pH (cf. Table 2). We then presented
a simple model to describe potential applied to electrokinetic
system electrodes in terms of potential drop across channel
lengths and several sources of potential losses. We decomposed
the applied potential into overpotential (using a Tafel formulation), equilibrium potential, the potential drop within endchannel reservoirs, and the potential drop over long, thin
channels. More practically, we performed three dimensional
simulations to estimate the effects of channel and reservoir
dimensions on the potential loss in reservoirs. We showed that in
a typical electrokinetic device with reservoirs of order 1 mL to
1 mL and characteristic channel dimensions below 100 mm, the
potential drop in the reservoirs is negligible for channels longer
than 1 cm. We also showed that non-homogeneous buffer
systems can increase potential drop in the reservoirs, and high
channel-to-reservoir conductivity ratios can yield reservoir
potential drops representing a significant fraction of applied
potential. We briefly discussed electrode materials and their
potential application to microfluidic electrokinetics. We also
suggested various methods for managing bubble generation and
lowering overpotential in electrokinetic systems.
Overall, the presentations serve as an introduction to the
principles associated with the coupling between electrochemistry,
acid–base equilibrium, and electromigration phenomena. We
argue that a basic understanding of this coupling is critical to the
design, analysis, testing, and optimization of electrokinetic
microfluidic systems.
Matthew Suss was supported by the FQRNT fellowship. We
gratefully acknowledge the support of the DARPA MTO
sponsored SPAWAR Grant N66001-09-1-2007. We thank
Thomas F. Jaramillo and Moran Bercovici for insightful
discussions and comments.
2468 | Lab Chip, 2009, 9, 2454–2469
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